Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems (Q612899)

The authors suggest a rigorous numerical proof yielding the existence of a transversal homoclinic orbit for a saddle point with higher dimensional unstable manifold. Their approach consists of the following three basic ingredients: {\parindent=6,5mm \begin{itemize}\item[(1)] Given the invertibility of a certain Jacobian matrix, a shadowing result guarantees the existence of such a homoclinic orbit near a suitable pseudo orbit. \item[(2)] A refinement procedure for numerically computing a pseudo-homoclinic orbit with sufficiently small local errors. \item[(3)] Verification that the homoclinic orbit is indeed transversal. \end{itemize}} As essential illustration for the above approach the authors use the family of three-dimensional Arneodo-Coullet-Tresser maps.